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<p>Fisher’s equation is a nonlinear partial differential equation that has significant applications in biology and related fields, motivating extensive research on its numerical solution. Although the numerical approximation of Fisher’s equation is well documented in the literature, this study presents a comprehensive literature review to highlight contemporary numerical methodologies. Additionally, a historical overview of Fisher’s reaction-diffusion equation highlights both the physical implications and the mathematical framework of its solutions. This investigation includes a systematic overview of the approximate numerical schemes existing in the literature and a comparative analysis of standard accuracy measures, such as the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({L_2}\)</EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\({L_\infty }\)</EquationSource> </InlineEquation> and the absolute error norms, which validate the quantitative evaluation of the method’s performance. In addition, graphical visualization of the approximate numerical solution is illustrated for some of the most common nonlinear FE types, which depicts the travelling-wave behaviour of the numerical solution. Overall, the review provides a consolidated assessment of existing numerical approaches and serves as a vital tool for future research on nonlinear reaction–diffusion models.</p>

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The Fisher’s Equation and its Numerical Solutions: A Comprehensive Methodical Review

  • 期刊论文

  • 发表日期：

    2026年7月30日

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  • Neelam Rana,
  • Neeraj Dhiman,
  • Mohammad Tamsir,
  • Robin Singh

摘要

Fisher’s equation is a nonlinear partial differential equation that has significant applications in biology and related fields, motivating extensive research on its numerical solution. Although the numerical approximation of Fisher’s equation is well documented in the literature, this study presents a comprehensive literature review to highlight contemporary numerical methodologies. Additionally, a historical overview of Fisher’s reaction-diffusion equation highlights both the physical implications and the mathematical framework of its solutions. This investigation includes a systematic overview of the approximate numerical schemes existing in the literature and a comparative analysis of standard accuracy measures, such as the \({L_2}\) , \({L_\infty }\) and the absolute error norms, which validate the quantitative evaluation of the method’s performance. In addition, graphical visualization of the approximate numerical solution is illustrated for some of the most common nonlinear FE types, which depicts the travelling-wave behaviour of the numerical solution. Overall, the review provides a consolidated assessment of existing numerical approaches and serves as a vital tool for future research on nonlinear reaction–diffusion models.